The longitudinal and lateral states of the decoupled linear dynamics are distinct sets of variables. They are conveniently remembered by the flight-control engineer by visualising the lateral dynamics through the cross section of the aircraft as shown in figure (a). The states are the associated velocities and angular displacements.
All the velocities which are shown vectorially in this figure This decoupling is illustrated in figure 6.1. (a) u, w, q are then states of the longitudinal dynamics. Additionally since there is only one rotational motion there is only the one associated angular displacement θ.
The velocities for the lateral dynamics p, v, r are then obtained by converting the translational velocities of figure 6.1 (a) into their assocated rotational velocities and vice versa as shown in figure (b).
There associated angular displacements φ and ψ. Clearly ψ which represents the heading cannot have an effect on the linear dyamics so this is neglected.
Noting that for small linear motions w can be converted to α, the states of the linearised longitudinal dynamics as shown in figure 6.1 (a) are u, w or α ,q and θ, and are affected by the elevator control δE. Since also for small linear motions v can be converted to β, the states of the linearised lateral dynamics as shown in figure 6.1 (a) are v or β, r ,p and φ, and are affected by the aileron δA and rudder δR controls.